gioko, a. (2007). eds ahl topic 9.3 electric field, potential and energy

Gioko, A. (2007). Eds

AHL AHL Topic Topic 99.3 .3

Electric Field, potential Electric Field, potential and Energyand Energy


Electric Potential EnergyElectric Potential EnergyIf you want to move a charge closer to a charged If you want to move a charge closer to a charged

sphere you have to push against the repulsive force sphere you have to push against the repulsive force You You do work do work and the charge and the charge gains gains electric electric

potential energy. potential energy. If you let go of the charge it will move away from If you let go of the charge it will move away from

the sphere, losing electric potential energy, but the sphere, losing electric potential energy, but gaining kinetic energy.gaining kinetic energy.


When you move a charge in an When you move a charge in an electric field its potential energy electric field its potential energy changes. changes.

This is like moving a mass in a This is like moving a mass in a gravitational field. gravitational field.


The The electric potential V electric potential V at any point in an at any point in an electric field is the electric field is the potential energy that each potential energy that each coulomb of positive charge would have if coulomb of positive charge would have if placed at that point in the field. placed at that point in the field.

The unit for electric potential is the joule per The unit for electric potential is the joule per coulomb (J Ccoulomb (J C‑1‑1), or the ), or the volt volt (V). (V).

Like gravitational potential it is a Like gravitational potential it is a scalar scalar quantity.quantity.


In the next figure, a charge In the next figure, a charge +q +q moves between points A moves between points A and B through a distance x in a uniform electric field. and B through a distance x in a uniform electric field.

The positive plate has a high potential and the negative The positive plate has a high potential and the negative plate a low potential. plate a low potential.

Positive charges of their own accord, move from a Positive charges of their own accord, move from a place of high electric potential to a place of low electric place of high electric potential to a place of low electric potential. potential.

Electrons move the other way, from low potential to Electrons move the other way, from low potential to high potential.high potential.


In moving from point A to point B in the In moving from point A to point B in the diagram, the positive charge diagram, the positive charge +q is +q is moving moving from a low electric potential to a high from a low electric potential to a high electric potential. electric potential.

The electric potential is therefore different The electric potential is therefore different at both points.at both points.


In order to move a charge from point A to point B, In order to move a charge from point A to point B, a force must be applied to the charge equal to a force must be applied to the charge equal to qE qE

(F = qE).(F = qE). force is applied through a distance x, then work force is applied through a distance x, then work

has to be done to move the charge, and there is an has to be done to move the charge, and there is an electric potential difference electric potential difference between the two between the two points. points.

work done is equivalent to the energy gained or work done is equivalent to the energy gained or lost in moving the charge through the electric field.lost in moving the charge through the electric field.


If a charge moves at an angle θ to an electric field, the component of the displacement parallel to the electric field is used as shown in the next figure.


Electric Potential DifferenceElectric Potential Difference

Potential difference Potential difference

What is difference in potential between two What is difference in potential between two points in an electric field points in an electric field

The potential difference or p.d. is the energy The potential difference or p.d. is the energy transferred when one coulomb of charge transferred when one coulomb of charge passes from one point to the other point.passes from one point to the other point.


The diagram shows some values of the electric potential at points The diagram shows some values of the electric potential at points in the electric field of a positively‑charged sphere in the electric field of a positively‑charged sphere

What is the p.d. between points A and B in the diagram?What is the p.d. between points A and B in the diagram?


Change in EnergyChange in Energy

Energy transferred, Energy transferred,

This could be This could be equal equal to the to the amountamount of electric of electric potentialpotential energy energy gainedgained or to the amount of or to the amount of kinetickinetic energy energy gained gained

W W = charge (q) x p.d.(V) = charge (q) x p.d.(V)

(joules)(joules) (coulombs) (coulombs) (volts) (volts)


The ElectronvoltThe ElectronvoltOne electron volt (1 eV) is defined as the energy One electron volt (1 eV) is defined as the energy

acquired by an electron as a result of moving acquired by an electron as a result of moving through a potential difference of one volt. through a potential difference of one volt. W = q x V W = q x V charge on an electron or proton is charge on an electron or proton is 1.6 x 101.6 x 10-19-19C C

Then Then W = W = 1.6 x 101.6 x 10-19-19CC x 1V x 1V W = 1.6 x 10W = 1.6 x 10-19 -19 J J

Therefore 1 eV = 1.6 x 10Therefore 1 eV = 1.6 x 10-19 -19 JJ


Electric Potential due to a Point ChargeElectric Potential due to a Point Charge

The electric potential at a point in an electric field is The electric potential at a point in an electric field is defined as being numerically equal to the work done defined as being numerically equal to the work done in bringing a unit positive charge from infinity to the in bringing a unit positive charge from infinity to the point. point.

Electric potential is a scalar quantity Electric potential is a scalar quantity and it has the and it has the volt V as its unit. volt V as its unit.

Based on this definition, the Based on this definition, the potential at infinity is potential at infinity is zerozero..


Consider a point r metres from a charged object. The potential at Consider a point r metres from a charged object. The potential at this point can be calculated using the followingthis point can be calculated using the following


Electric Field Strength and PotentialElectric Field Strength and Potential

Suppose that the charge Suppose that the charge +q is +q is moved a small moved a small distance by a force distance by a force F F from A to B so that from A to B so that the force can be the force can be considered considered constantconstant..

What is the work done?What is the work done?


The work done is given by: The work done is given by:

ΔΔW = Fx W = Fx ΔΔx x

The force The force F F and the electric field and the electric field E E are are oppositelyoppositely directed, and we know that: directed, and we know that:

F = ‑q x E F = ‑q x E

Therefore, the work done can be given as: Therefore, the work done can be given as:

ΔΔW = ‑qE x W = ‑qE x ΔΔ x = qV x = qVTherefore Therefore E = - E = - ΔΔV / V / ΔΔx x

This is theThis is the potential gradient. potential gradient.


Electric Field and Potential due to a charged Electric Field and Potential due to a charged spheresphere


In a charged sphere the charge distributes itself In a charged sphere the charge distributes itself evenly over the surfaceevenly over the surface. . Every part of the material of the conductor is at the Every part of the material of the conductor is at the same potentialsame potential. . Electric potential at a point is defined as Electric potential at a point is defined as

being numerically equal to the work done in bringing a unit positive being numerically equal to the work done in bringing a unit positive charge from infinity to that point, it has a constant value in every part charge from infinity to that point, it has a constant value in every part of the material of the conductorof the material of the conductor

potential is the potential is the same same at all points on the conducting surface, then at all points on the conducting surface, then

ΔΔ V / V / ΔΔx is zerox is zero. But . But E = ‑ E = ‑ ΔΔ V / V / ΔΔ x. x. The electric field inside the conductor is zero. There is no electric field The electric field inside the conductor is zero. There is no electric field

inside the conductorinside the conductor..


EquipotentialsEquipotentialsRegions in space where the electric potential of a Regions in space where the electric potential of a

charge distribution has a constant value are called charge distribution has a constant value are called equipotentials. equipotentials.

The places where the potential is constant in three The places where the potential is constant in three dimensions are called dimensions are called equipotential surfaces, equipotential surfaces, and and where they are constant in two dimensions they are where they are constant in two dimensions they are called called equipotential lines.equipotential lines.


They are in some ways analogous to the contour lines on They are in some ways analogous to the contour lines on topographic maps. topographic maps.

Similar also to gravitational potential. Similar also to gravitational potential.

In this case, the gravitational potential energy is constant as a In this case, the gravitational potential energy is constant as a mass moves around the contour lines because the mass mass moves around the contour lines because the mass remains at the same elevation above the earth's surface. remains at the same elevation above the earth's surface.

The gravitational field strength acts in a direction The gravitational field strength acts in a direction perpendicular to a contour line.perpendicular to a contour line.


Similarly, because the electric potential on an Similarly, because the electric potential on an equipotential line has the same value, no work can be equipotential line has the same value, no work can be done by an electric force when a test charge moves on an done by an electric force when a test charge moves on an equipotential. equipotential.

Therefore, the electric field cannot have a component Therefore, the electric field cannot have a component along an equipotential, and thus it must be everywhere along an equipotential, and thus it must be everywhere perpendicular to the equipotential surface or perpendicular to the equipotential surface or equipotential line. equipotential line.

This fact makes it easy to plot equipotentials if the lines of This fact makes it easy to plot equipotentials if the lines of force or lines of electric flux of an electric field are force or lines of electric flux of an electric field are known.known.


For example, there are a series of equipotential For example, there are a series of equipotential lines between two parallel plate conductors that are lines between two parallel plate conductors that are perpendicular to the electric field. perpendicular to the electric field.

There will be a series of concentric circles that map There will be a series of concentric circles that map out the equipotentials around an isolated positive out the equipotentials around an isolated positive sphere. sphere.

The lines of force and some equipotential lines for The lines of force and some equipotential lines for an isolated positive sphere are shown in the next an isolated positive sphere are shown in the next figures.figures.


Analogies exist between electric and gravitational Analogies exist between electric and gravitational fields.fields.

((a) Inverse square law of force a) Inverse square law of force Coulomb's law is similar in form to Newton's law of universal Coulomb's law is similar in form to Newton's law of universal

gravitation. gravitation. Both are inverse square laws with 1/(4Both are inverse square laws with 1/(4πεπε) in the electric case ) in the electric case

corresponding to the gravitational constant G. corresponding to the gravitational constant G. The main difference is that whilst electric forces can be attractive or The main difference is that whilst electric forces can be attractive or

repulsive, gravitational forces are always attractive. repulsive, gravitational forces are always attractive. Two types of electric charge are known but there is only one type of Two types of electric charge are known but there is only one type of

gravitational mass. gravitational mass. By comparison with electric forces, gravitational forces are extremely By comparison with electric forces, gravitational forces are extremely

weak.weak.


(b) Field strength (b) Field strength The field strength at a point in a gravitational field is defined as the The field strength at a point in a gravitational field is defined as the

force acting per unit mass placed at the point. force acting per unit mass placed at the point. Thus if a mass Thus if a mass m m in kilograms experiences a force F in newtons at a in kilograms experiences a force F in newtons at a

certain point in the earth's field, the strength of the field at that point certain point in the earth's field, the strength of the field at that point will be will be F/m F/m in newtons per kilogram. in newtons per kilogram.

This is also the acceleration This is also the acceleration a a the mass would have in metres per the mass would have in metres per second squared if it fell freely under gravity at this point (since F = second squared if it fell freely under gravity at this point (since F = ma)ma). .

The gravitational field strength and the acceleration due to gravity at a The gravitational field strength and the acceleration due to gravity at a point thus have the same value (i.e. point thus have the same value (i.e. F/m) F/m) and the same symbol, g, is and the same symbol, g, is used for both. At the earth's surface g = 9.8 N kgused for both. At the earth's surface g = 9.8 N kg‑‑' = 9.8 m s' = 9.8 m s‑2‑2 (vertically downwards).(vertically downwards).


(c) Field lines and equipotentials (c) Field lines and equipotentials These can also be drawn to represent gravitational fields but These can also be drawn to represent gravitational fields but

such fields are so weak, even near massive bodies, that there such fields are so weak, even near massive bodies, that there is no method of plotting field lines similar to those used for is no method of plotting field lines similar to those used for electric (and magnetic) fields. electric (and magnetic) fields.

Field lines for the earth are directed towards its centre and Field lines for the earth are directed towards its centre and the field is spherically symmetrical. the field is spherically symmetrical.

Over a small part of the earth's surface the field can be Over a small part of the earth's surface the field can be considered uniform, the lines being vertical, parallel and considered uniform, the lines being vertical, parallel and evenly spacedevenly spaced. .


(d) Potential and p.d(d) Potential and p.d. . Electric potentials and pds are measured in joules Electric potentials and pds are measured in joules

per coulomb (J Cper coulomb (J C‑1‑1) or volts; ) or volts; gravitational potentials and pds are measured in gravitational potentials and pds are measured in

joules per kilogram (J kgjoules per kilogram (J kg‑1‑1). ). As a mass moves away from the earth the As a mass moves away from the earth the

potential energy of the earth‑mass system potential energy of the earth‑mass system increases, transfer of energy from some other increases, transfer of energy from some other source being necessary.source being necessary.


If infinity is taken as the zero of gravitational If infinity is taken as the zero of gravitational potential (i.e. a point well out in space where no potential (i.e. a point well out in space where no more energy is needed for the mass to move more energy is needed for the mass to move further away from the earth) further away from the earth)

then the potential energy of the system will have then the potential energy of the system will have a negative value except when the mass is at a negative value except when the mass is at infinity. infinity.

At every point in the earth's field the potential is At every point in the earth's field the potential is therefore negative (see expression below), a fact therefore negative (see expression below), a fact which is characteristic of fields that exert which is characteristic of fields that exert attractive forces.attractive forces.


0bjectives covered• 9.3.1 Define electric potential and electric potential energy.• 9.3.2 State and apply the expression for electric potential due to a

point charge.• 9.3.3 State and apply the formula relating electric field strength

to electric potential gradient.• 9.3.4 Determine the potential due to one or more point charges.• 9.3.5 Describe and sketch the pattern of equipotential surfaces

due to one and two point charges.• 9.3.6 State the relation between equipotential surfaces and

electric field lines.• 9.3.7 Solve problems involving electric potential energy and electric

potential.


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gioko, a. (2007). eds ahl topic 9.3 electric field, potential and energy

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